## 4. Behavior of the continuum polarizationWe applied the computer code to the nine different model atmospheres introduced in Sect. 2.4. After a presentation of the resulting continuum polarization, we identify the reasons for the wavelength dependence (Sect. 4.1) and for the differences between the various model atmospheres (Sect. 4.2). The scattering coefficient and the temperature gradient turn out to be the most important physical quantities. Fig. 4 presents the calculated continuum polarization for different
model atmospheres as a function of -
The CLV is largely determined by simple geometry since Rayleigh and Thomson scattering act as dipole scattering (cf. Sect. 5.1). Due to axial symmetry the scattering polarization vanishes at disk center for all models. -
With increasing wavelength the polarization decreases steeply, mainly due to the wavelength dependence of the Rayleigh scattering. In Sect. 4.1.2 we will show a further effect due to the wavelength dependence of the Planck function. -
Within each of the model groups {AYFLUXT _{1}, AYP_{2}, AYCOOL_{8}} and {FALP_{3}, FALF_{4}, FALC_{5}, FALA_{7}} the polarization is smaller for hotter atmospheres. -
For the two model atmospheres with no chromosphere, AYCOOL _{8}and AND_{9}, the polarizations are similar to those of the other models. Thus the chromosphere does not seem to be very important for the formation of the polarization. AYFLUXT_{1}and AYP_{2}are the exceptions to this rule and have small contributions in the chromosphere, as will be shown in Sect. 4.2. -
The two average quiet Sun model atmospheres, FALC _{5}and MACKKL_{6}, differ significantly only in the upper chromosphere. However, their polarizations are almost identical, which again demonstrates the low relevance of the upper chromosphere.
## 4.1. Wavelength dependenceThis section is devoted to a qualitative study of the wavelength dependence of the continuum polarization. The essential points are summarized in Fig. 5. The results obtained below are valid for all models. The following discussion is divided into two parts corresponding to the two most important physical quantities, namely the scattering coefficient and the temperature gradient.
## 4.1.1. Scattering coefficientBetween 4000 Å and 8000 Å the scattering
coefficient decreases by approximately a factor of ten in the
photosphere, as shown in panels ## 4.1.2. Temperature gradientThe temperature gradient is directly responsible for limb
darkening. Panels
The greater the limb darkening the more anisotropic is the radiation field, and the greater the polarization produced. Fig. 5 clearly shows that the limb darkening decreases with increasing wavelength. This enhances the effect that the scattering coefficient yields a smaller polarization at larger wavelengths. It is interesting to note the fact that the wavelength dependence of the limb darkening can at least partially be explained by properties of the Planck function, as was pointed out to us by S.K. Solanki (private communication). For simplicity, we assume the absorption coefficient to be wavelength independent and the continuum intensity to be black-body radiation. We consider the Planck function and fix two temperatures and with . The ratio between two Planck functions, one at temperature , the other at , is given by which has the asymptotic values The ratio is a monotonically
decreasing function of wavelength if
, as shown in panel In a grey atmosphere the relation between temperature and optical
depth is wavelength independent. Therefore the lower value of
at higher wavelengths corresponds to
a less pronounced limb darkening, in agreement with panels ## 4.2. Sources of model dependenceIn this section the reasons for the model dependence of the
continuum polarization are investigated. Due to the form of the total
source function (5) it is natural to examine the influence of the
This however turns out to be insignificant for explaining the model dependence of the continuum polarization. Rather we find that the temperature gradient in combination with the scattering coefficient, , appear to be most important, as demonstrated in Fig. 6. Let us now discuss the various quantities plotted in that figure. ## 4.2.1. Contribution functionsFor diagnostic work it is useful to know the location in the
atmosphere where the emerging radiation is produced. This information
is contained in the The integration bounds in Eq. (14) are so chosen for formal convenience. However, the errors produced by integrating from minus to plus infinity, instead of only integrating over the atmospheric slab considered, are negligible. Panel Note that according to the definition (14) the contribution
function is proportional to
and not to
. Therefore,
is the relevant quantity when
interpreting the contribution functions (cf. panels ## 4.2.2. Relative scattering coefficientIn the photosphere the values of
are very similar in all nine model atmospheres (Fig. 6 panel f).
Therefore will not cause differences
in the polarization between the atmospheres in which Stokes ## 4.2.3. Scattering coefficientIn the photosphere the scattering coefficients are also almost
identical in all the model atmospheres and thus do not lead to a model
dependence. Only in the cases of AYFLUXT ## 4.2.4. Temperature gradientAccording to panel Let us now consider the model atmospheres FALF © European Southern Observatory (ESO) 1999 Online publication: December 16, 1998 |